Among other things, does the “random fashion” have to have a stopping point? If the chef picks randomly from the 4 condiment jars but does it 10 times, the chances of getting “one with everything” are pretty high. If the chef can pick one, two, three, or four times, nothing is stopping her from picking a fifth time.

And, of course, if she is picking randomly, Mike’s 15 events are not equally likely. She might well pick catsup or pickles 4 times out of 4. If she won’t pick catsup again because she has already picked it once, she is not picking randomly.

Again, if she is picking randomly, it matters how easy it is to pick each condiment. If there are 20 equally accessible condiment dispensers and 12 hold catsup, 5 hold pickles, 2 hold pickles, and 1 holds onions, the chances of getting catsup on any one pick are going to be 12 times greater than the chances of getting onions.

A burger joint has 4 condiments to add to its burgers: catsup, mustard, pickles, and onions. It offers burgers with none, one, two, three, or four of the condiments. If the joint prepares equal numbers of each kind and then hands the burgers out randomly, what is the the probability that a customer will get one of the following: catsup and onions, mustard and pickles, or one with everything?

That may well be what the question-asker is looking for. But it is not what was asked. Being able to translate the question shows that the student is becoming good at getting into the question-asker’s head. It does not show that the student is understanding random processes.

I think she misses some deeper issues. She can tell if a book is grammatically correct, if it makes logical errors, or gives wrong answers to problems. She knows what, by her standards, is a “quality” product. She makes a good case that lots of books that are sold today are not quality products. However, neither she nor anyone else knows which math books are most helpful to students, or even if one book is good for one group of students and a different book good for a different group. And that, after all, is what really matters.

It’s sad but true, but the easiest guide you can use is that, the more modern the K-12 textbook, the worse it’s going to be. Modern textbooks are so concerned with being PC that they often forget to even try to teach the core content anymore…

I agree the question is worded a little poorly, but I have seen a lot worse.
It appears to be a straight forward problem on independent trials:
(1) Catsup (50%) or no catsup (50%)
(2) Mustard (50%) or no mustard (50%)
(3) Pickles (50%) or no pickles (50%)
(4) Onions (50%) or no onions (50%)

The problem is equivalent to flipping a fair coin 4 times, and asking what is the probability that you obtain HTTH (where order matters).

That’s probably what the question-asker wants you to do. But that requires you to read a number of things into the question. It requires you to say, in effect, “I just learned to find probabilities for the flipping of a fair coin. I will assume this is just like that and use the same algorithm I used for coins.”

This is going to sound harsh but it requires you to NOT think deeply about the problem. It requires you to pretend that you can mechanically apply something because you have just learned it. It says, “I have just given you a hammer. Now pretend that everything you see here is a nail.” That bothers me.

Seems pretty clear to me, and I interpret it the same way jab does. The writing could be clearer, but the hamburger maker essentially flips a fair coin to decide on ketchup, flips again for mustard, flips again for pickles, flips again for onions. So each condiment choice is independent of each other condiment choice.

That means every four-condiment selection is equally likely. There are 16 (2*2*2*2) possible condiment selections. So a particular condiment selection has probability 1/16. You have a 1/16 chance of getting ketchup and onions only, 1/16 of getting mustard and pickles only, and 1/16 chance of getting everything. Adding those up, you’ve got a 3/16 chance of getting a burger with ketchup and onion only OR mustard and pickles only OR everything.

But if the question were more complicated, then enumerating the possibilities would be even more error prone. Suppose you’re getting a random sandwich at Sandwich Express (“Have It Our Way”). The possibilities ar white, whole wheat or rye bread; egg salad, ham, corned beef or roast beef; swiss cheese , American cheese or no cheese; mustard or no mustard; lettuce or no lettuce. You can’t order what you want; the servers make all the choices randomly with equal probability.

You ask for a sandwich. You want egg salad on white with mustard, lettuce and no cheese. What’s the chance you get it? I can compute the answer to that question in thirty seconds, the amount of time it takes me to figure how many choices there are for each element and multiply them together. If you solve the problem by enumeration, it’ll take you a while.

First, I am not talking about enumeration. I’m talking about mathematical counting. You seem like a smart fellow. I’m sure you took combinatorics.

Second, the problem you gave is structurally a different sort of problem, and therefore doesn’t require the same sort of work, and shouldn’t take any more than about 9 seconds, without writing anything down. 3 x 4 x 3 x 2 x 2? Please, we could both do that in our sleep.

Of course I’ve studied combinatorics. I see. You figured out, what’s four choose one (4), what’s four choose two (6), what’s four choose three (4), what’s four choose four (1), then you added them up. I find that a little indirect, but obviously it works, if you remember to also consider four choose zero.

Of course, the only REAL correct answer is that you’d refuse to eat the burger, because livestock hurts the Earth. You’d have a tofu burger instead. To answer otherwise these days is risking being denied your student loans, and ending up on some ‘watch lists’!

Yes, the question is poorly written, but being able to figure out the thought behind an unclear question, in particular a technical/quantitative one, is actually an important job skill. Seriously–I remember when I first graduated from college I struggled for a few years because real world problems and job specifications so often include information that is incomplete or simply wrong.

4 times 3 times 2 equals 24 possible results. All things being equal, it’s simply a numerical permutation. There are 24 possible ways to serve a hamburger limited to four codiments; without tasing the same combination twice.

This interpretation makes no sense if you look at what you’re supposed to answer. If all you’re concerned about is the order of the condiments, then how do you determine a probability for the mustard and pickles hamburger? In your scheme, all the burgers have mustard and pickles.

And what would the order of the condiments even mean? How would I even tell the difference between a hamburger with ketchup first and mustard second, versus a hamburger with mustard first and ketchup second?

The question is unanswerable without making assumptions about what “at random” means. It could be a 50/50 coin flip for each condiment, but it doesn’t have to be. Randomness is almost useless unless they specify its distribution.

I don’t believe that any complicating factors apply. The problem is pretty straightforward. There are 16 possible combinations of condiments, so each specific combo has a 1 in 16 chance of being served.